# Area under a function

Consider a continuous, positive function f: R --> R. The graph of f(x) is the set of points in the (x,y) Cartesian plane such that y = f(x).

See Attached file for Figure and questions.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

Please see the attachment.

We kow that the area can be represented as a definite integral as follows.

.

(a) Since , , then for any , we have . Then we have

and

Thus

(b) Set . From (a), we get . Since is a continuous function from the graph, then can reach maximum and minimum in the closed interval . Then we can find some , such that and . By the Intermediate Value Theorem, we can find some between and , such that . Then we have , where is between and , hence is in .

(c) We note that represent the area between and , according to (b), we can find some , such that . Therefore, we get .

(d) From part (c), we have , then we get

Let , then and the right side is the definition of . Then we have

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